$L^p$-solvability of the Poisson-Dirichlet problem and its applications to the regularity problem

Mihalis Mourgoglou, Bruno Poggi, Xavier Tolsa

Published 2022-07-21Version 1

We prove that the $L^{p'}$-solvability of the homogeneous Dirichlet problem for an elliptic operator $L=-\operatorname{div} A\nabla$ with real and merely bounded coefficients is equivalent to the $L^{p'}$-solvability of the Poisson Dirichlet problem $Lw=H-\operatorname{div} F$, assuming that $H$ and $F$ lie in certain Carleson-type spaces, and that the domain $\Omega\subset\mathbb R^{n+1}$, $n\geq2$, satisfies the corkscrew condition and has $n$-Ahlfors regular boundary. The $L^{p'}$-solvability of the Poisson problem (with an $L^{p'}$ estimate on the non-tangential maximal function) is new even when $L=-\Delta$ and $\Omega$ is the unit ball. In turn, we use this result to show that, in a bounded domain with uniformly $n$-rectifiable boundary that satisfies the corkscrew condition, $L^{p'}$-solvability of the homogeneous Dirichlet problem for an operator $L=-\operatorname{div} A\nabla$ satisfying the Dahlberg-Kenig-Pipher condition (of arbitrarily large constant) implies solvability of the $L^p$-regularity problem for the adjoint operator $L^*=-\operatorname{div} A^T \nabla$, where $1/p+1/p'=1$ and $A^T$ is the transpose matrix of $A$.